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user 10059
5
votes
$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential
And this is another construction.
Let $\sigma(x)=\exp(x)-1$ From this paper http://arxiv.org/abs/0812.4047 we know that
$$\exp(\sigma^{[p]}(t))=\sum_{n=0}^{\infty}B_n^p\frac{t^n}{n!}$$
where $B_n^p$ a …
- 10.1k
5
votes
$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential
You can find the half-iterate of a function from known integer iterates by using Newton series, for example:
$$f^{[1/2]}(x)=\sum_{m=0}^{\infty} \binom {1/2}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{[k]}(x) …
- 10.1k